What is snapping in Blender and how to use it properly?
Snapping is when you use the position or even orientation of something else in some operation to align/snap/move to/scale to/rotate to something that you are editing with that operation.
Blender has loads of various great snapping functionality.
Snap Menu
There is this snap menu that you can call with Shift+S:

Snapping in Transform Operations
And then there is also snapping in transform operations that depends on all these settings of your Transform Orientation and constraining of the transform operations or Axis Locking, Transform Pivot Point and Snapping Settings. These work in so many ways, that it would be hard to describe them all. You can find out more about it in the manual that you are reading and the links to it that I put in this answer, but the basic idea is that you combine all of those settings to make snapping situation that you happen to need. For example if I want to snap the center of Suzane's eye to the vertex of the point of the cone in my scene, one of the many ways to do that could be to set the Transform Pivot to the 3d Cursor, Snapping to Vertices and Snapping target to Center, this way, I could drag the cursor with Shift+ right-click, hold Ctrl down to enable snapping for it and snap it to the vertex on the eye, and then use grab G transform operation to snap the place the cursor was at to the vertex I hover my mouse over:

But you can see that there are many settings there and they will all work differently in combination with each other, you can also use rotation and scale transforms with various ways of snapping, and also you can use a to snap to multiple targets(you need to temporarily release Ctrl to press a if you are using it to only temporarily enable snapping during transform operation)
You will eventually get the feeling of what you need when you get familiar with all the settings, you should just try them out, and of course you will find it very well documented in the manual.